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In multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context.

To illustrate, consider three-dimensional Euclidean space, letting:

$\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}$
$\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k}$

be two vectors where i, j, k (also denoted e1, e2, e3) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum:

$\begin{array}{llll} \mathbf{ab} = & a_1 b_1 \mathbf{i i} & + a_1 b_2 \mathbf{i j} & + a_1 b_3 \mathbf{i k} \\ &+ a_2 b_1 \mathbf{j i} & + a_2 b_2 \mathbf{j j} & + a_2 b_3 \mathbf{j k}\\ &+ a_3 b_1 \mathbf{k i} & + a_3 b_2 \mathbf{k j} & + a_3 b_3 \mathbf{k k} \end{array}$

or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b):

$\mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}.$ This page uses content from Wikipedia. The original article was at Dyadics.The list of authors can be seen in the page history. As with the Math Wiki, the text of Wikipedia is available under the Creative Commons Licence.
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